Transfinite methods in geometry
A. Beutelspacher
P. J. Cameron
Dedicated to J. A. Thas on his fiftieth birthday
Abstract
Some very difficult questions in finite geometry (for example, concerning
ovoids, spreads, and extensions) have fairly trivial resolutions in the infinite
case, by standard transfinite induction. We give some old and new results of
this kind, and a number of open questions. The paper is expository in nature,
and is intended to illustrate the technique rather than give the most general
applications.
1
Introduction
Jef Thas has contributed to almost every part of finite geometry. Among these are
the related topics of ovoids, spreads and extensions. To cite just a couple among
many important papers, we mention [12], [13]. It is our purpose here to show that,
for infinite geometries, Jef’s great ingenuity is not needed; constructions are easy,
and everyone can have a go.
The basic tool is transfinite induction, which is outlined briefly in Section 2. In
the following sections, we construct extensions of Steiner systems and projective
planes; blocking sets, ovoids, and parallelisms of projective spaces; and partitions
into ovoids for a variety of quadrangle-like structures, leading to flat geometries with
diagrams like C2.C2 and “large sets” of Steiner systems.
The paper also contains a number of open questions.
Received by the editors in February 1994
AMS Mathematics Subject Classification: Primary 51A05, Secondary 05B25
Keywords: transfinite methods, ovoids, spreads, extensions, blocking sets.
Bull. Belg. Math. Soc. 3 (1994), 337–347
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2
A. Beutelspacher, P. J. Cameron
Transfinite induction
We work in Zermelo–Fraenkel set theory with the Axiom of Choice. This axiom will
normally be invoked in the form that
every set can be well-ordered (i.e., put into bijective correspondence with
an ordinal number).
Recall that a cardinal number is an ordinal which is not bijective with any smaller
ordinal; thus the smallest ordinal bijective with any set is a cardinal, and if |X| = n,
then we can well-order X as X = (xi : i < n), where the indices are ordinals.
We customarily divide ordinals into three classes: zero, successor ordinals (of
the form m + 1 for some ordinal m), and limit ordinals (the others). Any infinite
cardinal is a limit ordinal.
Transfinite induction is the principle asserting that
if P is a proposition about ordinals such that the truth of P (m) for all
m < n implies the truth of P (n), then P holds for all ordinals.
Often the hypothesis of the principle is separated into three parts (corresponding to
the division above). This is the form in which we will invoke it.
Theorem 2.1 Let P be a proposition about ordinal numbers. Then P holds for all
ordinals provided that the following conditions are satisfied:
(a) P (0) is true;
(b) P (m) implies P (m + 1) for any m;
(c) if n is a limit ordinal and P (m) holds for all m < n, then P (n) is true.
There is an analogous statement involving constructions rather than proofs (sometimes referred to as transfinite recursion). Typically, we start with the empty set;
at each successor ordinal, we add a point; and at a limit ordinal, we take the union
of the sets previously constructed. The possibility of adding a point at stage i often
requires a small argument depending on the fact that the number of points already
added is less than the total number of points available. Several examples below will
illustrate this.
3
Extensions of Steiner systems
As usual, a Steiner system S(t, k, v) consists of a v-set of points with a collection of
k-subsets called blocks, so that any t points lie in a unique block. In this paper, v
will always be an infinite cardinal number, and k may be finite or infinite; but we
assume that t is finite. In the finite case, non-degeneracy is forced by requiring that
0 < t < k < v. However, if v is infinite, there can be non-trivial systems with k = v;
we require that each block is a proper subset of the point set.
We begin with a simple fact about infinite Steiner systems.
Transfinite methods in geometry
339
Lemma 3.1 A S(t, k, v), with t ≥ 2 and v infinite, has v blocks.
Proof. There are not more than v blocks, since there are just v t-subsets of the
point set. If k < v, the assertion follows by cardinal arithmetic; so suppose that
k = v. Let B be a block, and S a (t − 1)-set disjoint from B. Then each point of B
lies in a block with S, and such blocks meet B in at most t − 1 points; so there are
v of them.
2
By an extension of a class K of incidence structures, we mean a structure with
the property that, for any point p, the derived structure at p (formed by the points
different from p and the blocks incident with p) is a member of K. An extension of
a particular member X of K additionally has the property that at least one such
substructure is isomorphic to X. An r-fold extension is defined analogously.
Thus, for example, an extension of the class of S(t, k, v) systems is a S(t + 1, k +
1, v + 1) system, while an extension of a particular Steiner system X must have
a derived structure isomorphic to X. Our first easy result is well-known, at least
for finite k. We give the proof in some detail; similar arguments later will just be
outlined.
Theorem 3.1 Any S(t, k, v) with t ≥ 2 and v infinite has an extension.
Proof. In order to extend an incidence structure X = (P, B), we add a point ∞,
and decree that the blocks containing ∞ in the extension are all sets B ∪ {∞} for
B ∈ B. The non-trivial part is the choice of the set C of blocks not containing
∞. In this case, we require that C is a collection of (k + 1)-subsets of P with the
properties that no (t + 1)-subset of a member of C lies also in a block of X, but
any (t + 1)-set not in a block of X is contained in a unique member of C. Such a
set is constructed by transfinite induction. We well-order the (t + 1)-subsets of P
which are not contained in members of X as (Ui : i < v). At the ith stage in the
construction, we must add to C a (k + 1)-set containing Ui , if one does not exist
already; it must meet the sets of B and the previously-chosen sets of C in at most t
points. We need a “technical lemma” to guarantee that such a set exists. For the
remainder of the proof, we assume that k is infinite; only trivial modifications are
required if k is finite.
Lemma 3.2 Let B be a set of k-subsets of the v-set P , with |B| = v. Suppose that
any t + 1 points of P lie in at most one member of B. If B 0 ⊆ B with |B 0| < v, then
there is a point of P lying in no set in B 0.
S
Proof. If k < v, then | B 0| ≤ k|B 0| < v; so assume that k = v. Take B ∈ B \ B 0.
Any member of B 0 contains at most t points of B; so some point lies in no such set.
2
Now we proceed to the construction of C. We assume that, at the conclusion of
stage i of the induction, we have a set Ci such that
(a) any two members of B ∪ Ci meet in at most t points;
(b) for j < i, Uj is contained in a (unique) member of Ci .
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A. Beutelspacher, P. J. Cameron
We start with C0 = ∅.
Suppose that Ci has been constructed. If Ui lies in a member of Ci , then set
Ci+1 = Ci . Otherwise, we must enlarge Ui to a k-set S meeting every member of
B ∪ Ci in at most t points. This is also done by transfinite induction; it is enough
to show that, if |S| < k, then a point can be added to S without violating this
condition. Let |S| = l < k. There are l t-subsets of S (if l is infinite; the argument
is similar if l is finite). Each lies in a unique member of B. Also, there are at most |i|
sets in Ci . So fewer than v sets meet S in t points, and by Lemma 3.2 we can choose
a point in none of them. Having constructed our k-set S, we put Ci+1 = Ci ∪ {S}.
S
If i is a limit ordinal, we put Ci = j<i Cj .
2
Now C = Cv is the required set of blocks in the extension.
4
Extensions of projective planes
An r-fold extension of an infinite projective plane is a Steiner system S(r + 2, v, v)
having the property that no two blocks meet in exactly r points.
Theorem 4.1 Let X = (P, B) be an r-fold extension of an infinite projective plane,
with r ≥ 0. Then X can be extended to a structure in which any two blocks not
containing the added point meet in exactly r + 2 points.
In particular,
any infinite projective plane has an r-fold extension in which any two
blocks meet in r + 1 points.
The proof is by transfinite induction as before. We require that a new block must
meet all existing blocks in r + 2 points. The possibility of doing this is guaranteed
by a technical lemma as follows.
Lemma 4.2 Let v and w be infinite cardinal numbers with w < v, and t a positive
integer. Let X be a set of size v, B a family of at most v subsets of X, each of size
v, with the property that any t points of X are contained in at most w members of
B, and any t + 1 points in at most one member of B. Let Y be a set of size less than
v with the property
(∀B ∈ B) |Y ∩ B| ≤ t.
Then there exists Z with Y ⊆ Z ⊆ X such that
(∀B ∈ B) |Z ∩ B| = t.
Proof. Enumerate B = (Bi : i < v). (We assume that |B| = v, though this
assumption is not necessary.) Set Y0 = Y , and define Yi for i ≤ v by transfinite
induction so that
(i) |Yi ∩ B| ≤ t for all B ∈ B;
(ii) |Yi ∩ Bj | = t for all j < i;
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341
(iii) |Yi | ≤ max{|Y |, |i|} if the right-hand side is infinite; otherwise |Yi | is finite.
These conditions clearly hold for i = 0.
Suppose that Yi is defined so that (i)–(iii) hold. Let |Yi ∩ Bi | = t − m. We want
to add m points of Bi to Yi to form Yi+1 without violating (i)–(iii). We describe
next how to add one point; repeat this procedure m times. (Of course, if m = 0,
there is nothing to do.)
There are fewer than v t-subsets of Yi , each lying in at most w members of B;
so fewer than v members of B meet Yi in t points. Each of these contains at most t
points of Bi ; so fewer than v points of Bi lie on a block with t points of Yi . Choose
a point p ∈ Bi different from all of these; we can add it to Yi without violating
(i)–(iii).
S
If i is a limit ordinal, set Yi = j<i Yj . Clearly (i) and (ii) hold, and |Yi | ≤
|i| max{|Y |, |i|} = max{|Y |, |i|}.
Now Z = Yv is the required set.
2
5
Blocking sets; extensions of projective spaces
A construction very similar to the one just given shows the following result.
Theorem 5.1 Let X be an infinite Steiner system S(t, v, v), and let l be a cardinal
number satisfying t ≤ l < v. Then the point set of X can be partitioned into subsets
(Pi : i < v) with the property that each set Pi intersects every block in exactly l
points.
Proof (outline). We proceed by induction as usual. Assume that the points are
well-ordered as (pj : j < v), and that by stage i all points pj for j < i, but not pi ,
lie in some set Pt . Assume also that the blocks are well-ordered as (Bk : k < v), and
that we have found a set S such that pi ∈ S, S is disjoint from the earlier sets Pj ,
and that S meets any block in at most l points, and all Bj for j < k in exactly l
points. We have to add at most l points of Bk to S, avoiding all intersections of Bk
with earlier sets Pj (at most |i| · l points), and all intersections with blocks B which
already meet S in l points (at most |S| of these if S is infinite, since l ≥ t); since
|Bk | = v, suitable points can be found one at a time.
2
The proposition applies to the lines of an infinite projective or affine plane, or a
projective or affine space over an infinite field, and yields blocking sets for these structures. We can block higher dimensional spaces, too. For example, Beutelspacher
and Mazzocca [2] showed the following proposition.
Theorem 5.2 If l, h, n are positive integers with h ≤ n and h + 2 ≤ l, and F an
infinite field, then there is an h-blocking set in PG(n, F ) or AG(n, F ), which meets
every h-subspace in at most l points and meets some h-subspace in exactly l points.
A consequence of Proposition 5.1 is the following extension result:
Theorem 5.3 Let F be an infinite field with v = |F |, and n a positive integer (at
least 2). Then there exists a S(3, v, v), all of whose derived structures are isomorphic
to the point-line structure of PG(n, F ).
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A. Beutelspacher, P. J. Cameron
Proof. Take a set S of points in PG(n + 1, F ) meeting every line in exactly two
points, and let the blocks be the intersections of S with all planes spanned by three
of its points.
2
This can be extended further. For example, a modification of the proof of Proposition 5.1 shows that, if n > 1, r > 0, and F is an infinite field, then there is a set of
points of PG(n+r, F ) meeting every r-subspace in a basis (a set of r +1 independent
points). The points and (r + 1)-space sections of such a set form an r-fold extension
of PG(n, F ).
6
Ovoids; extensions of affine spaces
Proposition 5.1, applied to the set of lines of a projective space PG(n, F ) over an
infinite field F , shows that we can choose a set of points meeting every line in exactly
k points, for any k ≥ 2. These sets are analogues of hyperovals in projective planes.
With a little care, it is possible to construct similar analogues of ovals.
Theorem 6.1 Let n, k be integers at least 2, and F an infinite field. Then there is
a set S of points of PG(n, F ) such that
(a) |S ∩ L| = 0, 1 or k for any line L;
(b) for any point p ∈ S, the union of the lines L satisfying S ∩ L = {p} is a
hyperplane Hp .
Lemma 6.2 Let S be a set of fewer than |F | points of PG(n, F ), and U a proper
subspace disjoint from S. Then U is contained in a hyperplane disjoint from S.
Proof. If U 6= ∅, choose a hyperplane of the quotient space by U which is disjoint
from all points hU, pi/U (using induction on n). If U = ∅, choose a point p 6∈ S and
replace U by {p}.
2
Proof of Theorem 6.1. Our strategy is to ensure that |S ∩ L| = k for each line
L not contained in a previously-constructed “tangent hyperplane” Hp . Accordingly,
well-order the lines as (Li : i < v), where v = |F |. At stage i, we will have a set Si
of points, and a hyperplane Hp containing p for each p ∈ Si , such that
(i) Hp ∩ Si = {p};
(ii) |L ∩ Si | ≤ k for all lines L;
(iii) for j < i, we have |Lj ∩ Si | = 0, 1 or k, and, if Si ∩ Lj = {p}, then Lj ⊆ Hp .
We begin as usual with S0 = ∅.
Suppose that Si has been constructed. If Li is contained in Hp for some p ∈ Si ,
then take Si+1 = Si . Otherwise, suppose that |Si ∩ Li | = k − m. We have to add m
points of Li to Si . We must avoid any point lying on a line containing k points of Si ,
and the intersection of Li with Hp for any p ∈ Si ; there are fewer than v such “bad”
points. If x is not bad, Lemma 6.2 permits us to find a hyperplane Hx disjoint from
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Si and containing x but not Li . Add x to Si . Repeat this m times, and let Si+1 be
the resulting set.
At limit ordinals, take the union as usual.
The set S = Sv is the required set.
2
The set just constructed with k = 2 is an ovoid in PG(n, F ). If P is an ovoid,
and B the set of sections by planes spanned by three points of P , then (P, B) is an
extension of the point-line structure of AG(n, F ) (and all its derived structures are
isomorphic): compare Proposition 5.3.
7
Spreads and parallelisms
A spread is a family of (pairwise disjoint) blocks which partitions the point set of an
incidence structure. A parallelism or resolution is a partition of the block set into
spreads. Beutelspacher [1] showed the following for projective spaces.
Theorem 7.1 Let F be an infinite field and n, h positive integers. Let X be the
incidence structure of points and h-subspaces of PG(n, F ). Then the following are
equivalent:
(a) X has a spread;
(b) X has a parallelism;
(c) n ≥ 2h + 1.
Proof. The proof is by arguments of a type which should now be familiar. Note
that the condition n ≥ 2h + 1 is equivalent to the statement that there exist two
disjoint h-subspaces. The appropriate technical lemma generalizes Lemma 6.2 from
“points” and “hyperplanes” to “h-spaces” and “(n − h − 1)-spaces”. We will generalize this further in Section 9.
2
It is possible to modify the requirements on the spread or parallelism. To give
one example (Cameron [4]):
Theorem 7.1 Let F be an infinite field, and let S, T be sets of points and planes
respectively in PG(3, F ), with |S| + |T | < |F |. Then there exists a set L of pairwise
disjoint lines with the properties
(a) a point lies on a line of L if and only if it is not in S;
(b) a plane contains a line of L if and only if it is not in T .
For example, take S = ∅, T = {Π}. In the affine 3-space for which Π is the plane
at infinity, L is a spread containing one line from each parallel class.
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A. Beutelspacher, P. J. Cameron
Ovoids in other structures
This section summarises results which have appeared in Cameron [5]. In structures
like generalized polygons, a different definition of ovoids is normally used. So, in
this section, an ovoid in an incidence structure is a set of points which meets every
block in exactly one point. Thus an ovoid is the dual concept to a spread, and the
dual concept to a parallelism is a partition of the point set into ovoids.
Theorem 8.1 Let X = (P, B) be an incidence structure with v points, where v is
infinite. Suppose that any point lies on v blocks. Suppose further that, if B is a block
and S a set of fewer than v points outside B, then there are v points of B which
lie in a block with no point of S. Then the point set of X can be partitioned into
ovoids.
We call an incidence structure satisfying the hypotheses of the theorem standard.
Many incidence structures are standard: for example, any generalized quadrangle
with s = t infinite. (In a generalized quadrangle, or GQ, any point p not on a block
B is collinear with exactly one point of B; so, with B and S as in the theorem, at
most |S| points of B are collinear with a point of S.)
We note one contrast with the finite case here. Let X be a symplectic GQ over
F , embedded in PG(3, F ) in the usual way (as the points and totally isotropic lines
for an alternating form). If F is finite, then any ovoid of X is an ovoid (in a different
sense!) of PG(3, F ) (see Cameron [4]). However, if F is infinite, we can start the
inductive construction with more than two (and less than |F |) points of a hyperbolic
line, to obtain an ovoid of X which is not an ovoid of the projective space.
Finally, we give a brief application of Theorem 8.1 to diagram geometries (Buekenhout [3]). Suppose that X1 and X2 are two rank 2 geometries (i.e., incidence structures), each having a partition of the point set into ovoids. Suppose that the numbers
of ovoids in the partitions are the same for the two geometries. Then there is a rank
3 geometry with string diagram having residues X1∗ and X2 . Moreover, this geometry is flat: if the types are 0, 1, 2, then any two varieties of types 0 and 2 are
incident. Taking X1 and X2 to be GQs with s = t infinite, we obtain a huge number
of geometries with the diagram C2.C2 .
The construction proceeds as follows. Type 0 varieties are the blocks of X1 , and
type 2 varieties are the blocks of X2 ; and we declare any varieties of types 0 and 2
incident. We choose a bijection θ from the ovoids in the partition of X1 to those in
the partition of X2 . Now a type 1 variety is defined to be a pair (p1 , p2 ), where p1 is
a point of X1 , p2 a point of X2 , and the ovoids containing these points correspond
under θ. A block B1 of X1 is incident with the pair (p1 , p2 ) if and only if it is incident
with p1 , and similarly for X2 . The diagram properties are easily checked. In fact,
we can “glue together” any number of rank 2 geometries all of which have partitions
into the same number of ovoids. (This construction is due to Kantor [10].)
Any C2.C2 geometry has the property that its universal 2-cover is a building
(Tits [14]). So we obtain many strange C2 .C2 buildings.
Transfinite methods in geometry
9
345
Large sets of Steiner systems
A large set of Steiner systems S(t, k, v) is a partition of the set of all k-subsets of
a v-set into Steiner systems. In this section, as an application of Theorem 8.1, we
show the existence of large sets for t, k finite and v infinite. We also give a vector
space analogue, which extends a result of Ceccherini and Tallini [6].
Let X be an infinite set, with |X| = v. Suppose that t and k are finite with
t < k. Let I(t, k, v) be the incidence structure whose points and blocks are the
k-subsets and t-subsets of X, with incidence being reversed inclusion.
Theorem 9.1 I(t, k, v) is standard if t < k are finite and v is infinite.
Proof. The number of blocks (t-subsets of X) is equal to v; and the number of
points on a block T is the number of (k − t)-subsets of X \ T , which is also v.
Given a t-set T , and a set {Kj : j ∈ J } of fewer than v k-sets not containing
t, the union of all these sets has cardinality less than v, so there are v elements of
X not contained in T or any Kj . Now any k-set containing T and k − t of these
elements will have the required property. (Two k-sets are collinear if and only if
they meet in at least t points.)
2
Now an ovoid in I(t, k, v) is an S(t, k, v) by definition, and a partition into ovoids
is a large set of such Steiner systems.
We can play the same game with the incidence structure I = I(t, k, v, F ) of
k-subspaces and t-subspaces of V , where V = V (v, F ). Details will not be given
here.
Theorem 9.2 For t < k finite, I(t, k, v, F ) is standard if v ≥ 2k − t + 1 and either
v or F is infinite.
Remark. The bound v ≥ 2k−t+1 is clearly necessary for the existence of a “vector
space Steiner system”, since such a system must contain two k-spaces intersecting
in a (t − 1)-space. Note that we recover Theorem 7.1 by taking t = 1. In the case
where v = 2k − t + 1, any two k-spaces in the same system must intersect in a
(t − 1)-space. In particular, if t = 2 and v = 2k − 1, each system is a projective
plane.
10
Miscellanea
Shult and Thas [11] described the universal projective embeddings of dual orthogonal
spaces over a field F , under the assumption that the symplectic GQ over F has no
ovoids. They showed that the finite fields with this property are precisely those of
odd order. Theorem 8.1 implies that no infinite field has this property.
We have not touched here on the literature on free constructions of combinatorial
objects such as projective planes.
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A. Beutelspacher, P. J. Cameron
Problems
Question 11.1. Which pairs of structures have a common extension (i.e., are both
isomorphic to derived structures of the same structure)? A necessary condition is
that the two structures should have isomorphic derived structures. Note that any
two projective planes of the same order satisfy this condition.
Question 11.2. For which structures X is there a structure Y , all of whose derived
structures are isomorphic to X? When is there such a structure with a transitive
automorphism group?
Question 11.3. For which t, k, v is there a Steiner system S(t, k, v) with a ttransitive automorphism group? Note that the techniques of Cohen [7], [8] show
that, for any finite k > t, there exist (countable) Steiner systems with (t − 1)transitive groups. Moreover, the groups Cohen obtains are free of countable rank,
and the generators act as cyclic permutations. Perhaps more model-theoretic techniques like those of Hrushovski [9] will settle this question.
Question 11.4. When is there a set of points in PG(n, F ) or AG(n, F ) meeting
every h-space in exactly l points?
Question 11.5. Can the point set of PG(n, F ) be partitioned into ovoids?
Question 11.6. Can every infinite affine plane be extended (to an inversive plane)?
(See Thas [13] for a finite analogue.) What about further extensions?
Question 11.7. Which GQs have extensions? (For results on the finite version of
this question, see Thas [12].)
Question 11.8. For n > 2, does AG(n, F ) have a parallelism in which each parallel
class has just one line in common with each class of the “standard” parallelism? (Cf.
Proposition 7.1)
References
[1] A. Beutelspacher.
Parallelismen in unendlichen projektiven Räumen
endlicher Dimension. Geom. Dedicata, 7, pp. 499–506, 1978.
[2] A. Beutelspacher and F. Mazzocca. Blocking-sets in infinite projective
and affine spaces. J. Geometry, 28, pp. 111–116, 1987.
[3] F. Buekenhout. Diagrams for geometries and groups. J. Combin. Theory
Ser. A, 27, pp. 121–151, 1979.
[4] P. J. Cameron. Projective and Polar Spaces, volume 13 of QMW Maths
Notes. Queen Mary and Westfield College, University of London, 1992.
[5] P. J. Cameron. Ovoids in infinite incidence structures. Archiv Math., 62,
pp. 189–192, 1994.
[6] P. V. Ceccherini and G. Tallini. A new class of planar pi-spaces and some
related topics. J. Geometry, 27, pp. 69–86, 1986.
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[7] D. A. Cohen. An infinite perfect code with a just k-transitive automorphism
group, 1993. Preprint.
[8] D. A. Cohen. On the ubiquity of infinite perfect codes and infinite steiner
systems, 1993. Preprint.
[9] E. Hrushovski. A new strongly minimal set. Ann. Pure Appl. Logic, 62,
pp. 147–166, 1993.
[10] W. M. Kantor. Generalized polygons, SCABs and GABs. In L. A. Rosati, editor, Buildings and the Geometry of Diagrams, Proceedings Como 1984, volume
1181 of Lecture Notes in Math., pages 79–158. Springer Verlag, 1986.
[11] E. E. Shult and J. A. Thas. Hyperplanes of dual polar spaces and the spin
module. Arch. Math., 59, pp. 610–623, 1992.
[12] J. A. Thas. Extensions of finite generalized quadrangles. Symp. Math., 28,
pp. 127–143, 1986.
[13] J. A. Thas. Solution of a classical problem on finite inversive planes. In
W. M. Kantor et al., editors, Finite Geometries, Buildings and Related Topics,
pages 145–159. Oxford University Press, 1990.
[14] J. Tits. A local approach to buildings. In The Geometric Vein (Coxeter
Festschrift), pages 519–547. Springer Verlag, 1981.
A. Beutelspacher
Mathematisches Institut
Justus Liebig Universität
Arndtstr. 2
D-35392 Giessen
Germany
P. J. Cameron
School of Mathematical Sciences
Queen Mary and Westfield College
Mile End Road
London E1 4NS
U.K.